A number of places in this world have a lot of money and a lot of population, but no room. This tends to make the people miserable, as the rent increases continuously, everything gets crowded, and other problems occur because of the high population. Some of these places can no longer expand, because the surrounding land is owned by other entities. In particular, I think of New York City in the United States, whose surrounding land is owned by New Jersey, Yonkers, and Pennsylvania, and also Japan, an island nation with territorial arguments with Korea, China, and Russia. Japan especially suffers from crowding, since it has half the population of the United States, but is smaller than even the state of California in area. According to Wikipedia, California is 423,97 km^2, while Japan is merely 377,873 km².
For the crowded places on earth that have the money to deal with it, I suggest a land-filling policy, as most of these places are costal. The Netherlands have reclaimed 1/5th of their land from the ocean, so I know this is do-able. While it is possible to reclaim straight from the shore, many shore constructions were built because they were close to the shore, so to preserve land values, this technique should be used to create new islands.
After the site has been selected, workers should construct a caisson, a watertight pressurized tube, and install a pillar in the ocean. This should be done around the entire parameter of the would-be island. The watertight part can be removed when the pillars are fully constructed, and may or may not be recyclable, depending on how the construction and removal works. The pillars should all have notches facing each other. A large metal or plastic sheet of considerable thickness is placed between the notches, forming a water-tight seal. When the water is pumped from the center, this leaves a giant hole in the sea.
The hole should be filled with, at first, cheap materials. Especially garbage. Garbage should be readily available from any overcrowded area. Middle layers can even contain toxic waste between layers of cement. When the hole is almost filled, the top layer should be filled with only soil or compost, as this layer will directly be used by the inhabitants of the new island. It would also be wise to put a subway system, underground wiring, or water pipes, as appropriate, in this layer.
Finally, when the surface has been reached by the dirt layer, the island is avaialble for construction. Developers should move in and build large, high density apartments and businesses, and construct a means of transportation to the rest of the city, by bridge, subway, or ferry. Finally, everyone moves in, having more room to grow and build.
I recommend that New York build to the east or Northeast in the Atlantic to avoid the possibility of claims from New Jersey, and similarly recommend that Japan expand east into the Pacific to avoid conflicts with Korea.
I estimate this method to cost $5 million per square kilometer.